Solve 5^3n-2=125

$${ 5 }^{ 3n-2 } = 125$$

Solve for n

$$n = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$5^{3n-2}=125$$

Take the logarithm of both sides of the equation.

$$\log(5^{3n-2})=\log(125)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$\left(3n-2\right)\log(5)=\log(125)$$

Divide both sides by \log(5).

$$3n-2=\frac{\log(125)}{\log(5)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$3n-2=\log_{5}\left(125\right)$$

Add 2 to both sides of the equation.

$$3n=3-\left(-2\right)$$

Divide both sides by 3.

$$n=\frac{5}{3}$$